Question 236·Hard·Math

0 / 2000

Note: Figure not drawn to scale.

In the figure shown, $\overline{RS}$ and $\overline{MN}$ intersect at point $T$ . The angles at $M$ and $S$ are congruent (each marked $b^{\circ}$ ). $TM = 35$ , $TS = 25$ , $MR = 56$ , and $TN = 60$ . What is the length of $\overline{SN}$ ?

Your answer

Crossing segments and a pair of marked equal angles are a cue to look for AA similarity: two equal angle pairs make triangles the same shape, though not necessarily the same size. Find the opposite angles at the crossing, match vertices, and write matching side ratios in one direction. A Desmos regression can solve the proportion. The trap is pairing sides by where they appear instead of by their matching endpoints.

Hints

Hint 1
At a crossing, vertical angles sit opposite each other and are equal. The angle between $TM$ and $TR$ sits directly opposite the angle between $TS$ and $TN$ . What does that give you besides the marked angle pair?
Hint 2
In similar triangles, equal angles identify matching vertices. Pair the marked angles, then pair the angles at $T$ . Use those matches to find which side corresponds to $MR$ , rather than judging by the picture's layout.
Hint 3
A scale factor multiplies every matching side by the same amount. Compare $ST$ with $MT$ , then compare $SN$ with $MR$ in the same order. Which of those ratios contains the length you need?

Step-by-step

Match the angles, then solve a proportion
Step 1
Identify the vertical angles
Because the segments cross at $T$ , $\angle MTR$ and $\angle STN$ are vertical angles, the opposite angles at a crossing, so they're equal.
Step 2
Show that the triangles are similar
The marked equal angles pair $M$ with $S$ . By AA similarity, two equal angle pairs make the triangles similar: $\triangle MTR\sim\triangle STN$ . Their vertices match $M\leftrightarrow S$ , $T\leftrightarrow T$ , and $R\leftrightarrow N$ .
Step 3
Match sides and write the proportion
A corresponding side joins matching vertices, so $MT\leftrightarrow ST$ and $MR\leftrightarrow SN$ . Let $x=SN$ . Keep the second triangle over the first in both ratios: $\frac{x}{56}=\frac{25}{35}$ . Match sides by their angle endpoints before writing a ratio. Don't pair $TN=60$ with $MT$ : $TN$ matches $TR$ , not $MT$ .
Step 4
Find the requested length
Type $\frac{x_1}{56}\sim\frac{25}{35}$ in Desmos. Using $x_1$ instead of $x$ and $\sim$ instead of $=$ makes this a regression: Desmos finds the value that makes the ratios equal. Under PARAMETERS, it shows $x_1=40$ . Since $x$ stands for $SN$ , the length of $\overline{SN}$ is $40$ . Grid in 40.

Question 236·Hard·Math

Strategy

Hints

Step-by-step

Lessons that teach this

Study Hints

Strategy

Hints

Step-by-step

Lessons that teach this